\(\mathbf {Remark\;2.12}\) : In this way, at least formallyFor operators \(\pi _{\omega }(a)\)  either in the continuous or smooth
algebras, then the expression \(\mathcal {F}_{a}(\omega _{\underline{\psi }_{N_{1}N_{2}}})\doteq \omega _{\underline{\psi }_{N_{1}N_{2}}}(a)\doteq \left(\underline{\psi }_{N_{1}N_{2}},\pi _{\omega }(a)\underline{\psi }_{N_{1}N_{2}}\right)_{\mathcal {H}_{\omega }}\) 
does define a genuine pure algebraic state (recall that pure
states are in \(1-1\)  correspondence with the points \(x\)  of the spacetime
\(M\)  in the classical case and \(\mathcal {F}_{f}(\omega _{x})\doteq \omega _{x}(f)=f(x)\) 
is just the Gelfand transform [1]}, where \(\mathcal {F}_{f}\) 
corresponds to the continous function \(f\)  on \(M\) .), we can see \(\underline{\psi }_{N_{1}N_{2}}\)  as a “pure algebraic
state” \(\omega _{\underline{\psi }_{N_{1}N_{2}}}\)  which acts as
\(\omega _{\underline{\psi }_{N_{1}N_{2}}}(D_{P_{N_{1}}P_{N_{2}}}^{2})\doteq \left(\underline{\psi }_{N_{1}N_{2}},D_{P_{N_{1}}P_{N_{2}}}^{2}\underline{\psi }_{N_{1}N_{2}}\right)_{\mathcal {H}_{\omega }}.\) 
